Past

There has been much recent progress in extending Griffith's criterion for

crack growth into mathematical models for quasi-static crack evolution

that are well-posed, in the sense that there exist solutions that can be

numerically approximated. However, mathematical progress in dynamic

fracture (crack growth consistent with Griffith's criterion, together with

elastodynamics) has been meager. We describe some recent results on a

phase-field model of dynamic fracture, as well as some models based on a

"sharp interface" instead of a phase-field.

Some possible strategies for showing existence for these last models will

also be described.

Baumslag-Solitar groups are very simple groups which are not Hopfian (they are isomorphic to proper quotients). I will discuss these groups, as well as their obvious generalizations, with emphasis on their automorphisms and their generating sets

We will examine the typical structure of random polytopes by projecting the three fundamental regular polytopes: the simplex, cross-polytope, and hypercube. Along the way we will explore the implications of their structure for information acquisition and optimization. Examples of these implications include: that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n), or that for a surprisingly large set of optimization problems the feasible set is actually a point. These implications are driving a new signal processing paradigm, Compressed Sensing, which has already lead to substantive improvements in various imaging modalities. This work is joint with David L. Donoho.

We will consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We shall specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. Some limit theorems related to the distribution of the allelic partition will be also presented.

We discuss how various types of orientational and

translational ordering in different liquid crystal phases are

described by macroscopic tensor order parameters. In

particular, we consider a mean-field molecular-statistical

theory of the transition from the orthogonal uniaxial smectic

phase and the tilted biaxial phase composed of biaxial

molecules. The relationship between macroscopic order

parameters, molecular invariant tensors and the symmetry of

biaxial molecules is discussed in detail. Finally we use

microscopic and macroscopic symmetry arguments to consider the

mechanisms of the ferroelectric ordering in tilted smectic

phases determined by molecular chirality.

Abstract: There has been considerable interest recently in the relation between certain 3d supersymmetric Chern-Simons theories, M2-branes, and the AdS_4/CFT_3 correspondence. In this talk I will show that the moduli space of a 3d N=2 Chern-Simons quiver gauge theory always contains a certain branch of the moduli space of a parent 4d N=1 quiver gauge theory. In particular, starting with a 4d quiver theory dual to a Calabi-Yau 3-fold singularity, for certain general choices of Chern-Simons levels this branch of the corresponding 3d theory is a Calabi-Yau 4-fold singularity. This leads to a simple general method for constructing candidate 3d N=2 superconformal Chern-Simons quivers with AdS_4 gravity duals. As simple, but non-trivial, examples, I will identify a family of Chern-Simons quiver gauge theories which are candidate AdS_4/CFT_3 duals to an infinite class of toric Sasaki-Einstein seven-manifolds with explicit metrics.

We begin by the study of the problem of the exponential utility maximization. As opposed to most of the papers dealing with this subject, the investors’ trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. Instead of the well-known convex duality approach, we apply a backward stochastic differential equation (BSDE) approach. This leads to the study of quadratic BSDEs. The second part gives the recent result on the existence and uniqueness of solution to quadratic BSDEs. We give also the connection between these BSDEs and quadratic PDEs. The last part will show that quadratic BSDE is critic. That is, if the BSDE is superquadratic, there exists always some BSDE without solution; and there is infinite many solutions when there is one solution. This phenomenon does not exist for quadratic and superquadratic PDEs.

[This is a joint seminar with OASIS]

A formulation of quantum mechanics in terms of symmetric monoidal categories

provides a logical foundation as well as a purely diagrammatic calculus for

it. This approach was initiated in 2004 in a joint paper with Samson

Abramsky (Ox). An important role is played by certain Frobenius comonoids,

abstract bases in short, which provide an abstract account both on classical

data and on quantum superposition. Dusko Pavlovic (Ox), Jamie Vicary (Ox)

and I showed that these abstract bases are indeed in 1-1 correspondence with

bases in the category of Hilbert spaces, linear maps, and the tensor

product. There is a close relation between these abstract bases and linear

logic. Joint work with Ross Duncan (Ox) shows how incompatible abstract

basis interact; the resulting structures provide a both logical and

diagrammatic account which is sufficiently expressive to describe any state

and operation of "standard" quantum theory, and solve standard problems in a

non-standard manner, either by diagrammatic rewrite or by automation.

But are there interesting non-standard models too, and what do these teach

us? In this talk we will survey the above discussed approach, present some

non-standard models, and discuss in how they provide new insights in quantum

non-locality, which arguably caused the most striking paradigm shift of any

discovery in physics during the previous century. The latter is joint work

with Bill Edwards (Ox) and Rob Spekkens (Perimeter Institute).

In a paper from 1994, 'The density of rational points on non-singular hypersurfaces', Heath-Brown developed a `multi-dimensional q-analogue'

of van der Corput's method of exponential sums, giving good bounds for the density of solutions to Diophantine equations in many variables. I will discuss this method and present some generalizations.

We focus on domain decomposition methods for systems of PDEs (versus scalar PDEs). The Smith factorization (a "pure" algebra tool) is used systematically to derive new domain decompositions methods for symmetric and unsymmetric systems of PDEs: the compressible Euler equations, the Stokes and Oseen (linearized Navier-Stokes) problem. We will focus on the Stokes system. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem.

We study the asymptotics of the Stokes problem in cylinders becoming unbounded in the direction of their axis. We consider

especially the case where the forces are independent of the axis coordinate and the case where they are periodic along the axis, but the same

techniques also work in a more general framework.

We present in detail the case of constant forces (in the axial direction) since it is probably the most interesting for applications and also

because it allows to present the main ideas in the simplest way. Then we briefly present the case of periodic forces on general periodic domains. Finally, we give a result under much more general assumptions on the applied forces.

I shall discuss a number of problems to do with approximating the value function of an American Put option in the Black-Scholes model. This is essentially a variant of the oxygen-consumption problem, a parabolic free boundary (obstacle) problem which is closely related to the Stefan problem. Having reviewed the short-time behaviour from the perspective of ray theory, I shall focus on constructing approximations in the case when there is a discretely paid dividend yield.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk is the second of two parts.

'A two-point set is a subset of the plane which meets every line in exactly two points. The existence of two-point sets was shown by Mazurkiewicz in 1914, and the main open problem concerning these objects is to determine if there exist Borel two-point sets. If this question has a positive answer, then we most likely need to be able to construct a two-point set without making use of a well-ordering of the real line, as is currently the usual technique.

We discuss recent work by Robin Knight, Rolf Suabedissen and the speaker, and (independently) by Arnold Miller, which show that it is consistent with ZF that the real line cannot be well-ordered and also that two-point sets exist.'

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings, the system sticks to a permanent critical state (i.e. exhibits so-called self-organised critical behaviour). The talk will be based on joint work with Balint Toth.

We study non-commutative analogues of Serre-ï¬~Abrations in topology. We shall present several examples of such ï¬~Abrations and give applications for the computation of the K-theory of certain C*-algebras. (Joint work with Ryszard Nest and Herve Oyono-Oyono.)

Let M be a closed spin manifold.

Gromov and Lawson have shown that the presence of certain "enlargeable"

submanifolds of codimension 2 is an obstruction to the existence of a Riemannian metric with positive scalar curvature on M.

In joint work with Hanke, we refine the geoemtric condition of

"enlargeability": it suffices that a K-theoretic index obstruction of the submanifold doesn't vanish.

A "folk conjecture" asserts that all index type obstructions to positive scalar curvature should be read off from the corresponding index for the ambient manifold M (this this is equivalent to a small part of the strong Novikov conjecture). We address this question for the obstruction above and discuss partial results.

The uniform spanning forest (USF) in a graph

is a random spanning forest obtained as the limit of uniformly chosen spanning

trees on finite subgraphs. The USF is known to have stochastic dimension 4 on

graphs that are "at least 4 dimensional" in a certain sense. In this

talk I will look at more detailed estimates on the geometry of a fixed

component of the USF in the special case of the d-dimensional integer lattice,

d > 4. This is motivated in part by the study of random walk restricted to a

fixed component of the USF.

A random environment (in Z^d) is a collection of (random) transition probabilities, indexed by sites. Perform now a random walk using these transitions. This model is easy to describe, yet presents significant challenges to analysis. In particular, even elementary questions concerning long term behavior, such as the existence of a law of large numbers, are open. I will review in this talk the model, its history, and recent advance, focusing on examples of unexpected behavior.

Stress levels embedded in S&P 500 options are constructed and re-ported. The stress function used is MINMAXV AR: Seven joint laws for the top 50 stocks in the index are considered. The first time changes a Gaussian one factor copula. The remaining six employ correlated Brownian motion independently time changed in each coordinate. Four models use daily returns, either run as Lévy processes or scaled, to the option maturity. The last two employ risk neutral marginals from the V GSSD and CGMY SSD Sato processes. The smallest stress function uses CGMY SSD risk neutral marginals and Lévy correlation. Running the Lévy process yields a lower stress surface than scaling to the option maturity. Static hedging of basket options to a particular level of accept- ability is shown to substantially lower the price at which the basket option may be o¤ered.

"Stickelberger's famous theorem (from 1890) gives an explicit ideal which annihilates the imaginary part of the class group of an abelian field as a module for the group-ring of the Galois group. In the 1980s Tate and Brumer proposed a generalisation of Stickelberger's Theorem (and his ideal) to other abelian extensions of number fields, the so-called `Brumer-Stark conjecture'.

I shall discuss some of the many unresolved issues connected with the annihilation of class groups of number fields. For instance, should the (generalised) Stickelberger ideal be the full annihilator, the Fitting ideal or what? And what can we say in the plus part (where Stickelberger's Theorem is trivial)?"

In this talk we present a new construction of a Wedderburn basis for

the generic q-Schur algebra using the Du-Kazhdan-Lusztig basis. We show

that this gives rise to a new view on the Du-Lusztig homomorphism to the

asymptotic algebra. At the end we explain a potential plan for an attack

on James' conjecture using a reformulation by Meinolf Geck.

The talk starts with a gentle recollection of facts about

Iwahori-Hecke-Algebras of type A and q-Schur algebras and aims to be

accessible to people who are not (yet) experts in the representation

theory of q-Schur algebras.

All this is joint work with Olivier Brunat (Bochum).

The butterfly-shaped Lorenz attractor is a fractal set made up of infinitely many periodic orbits. Ever since Lorenz (1963) introduced a system of three simple ordinary differential equations, much of the discussion of his system and its strange attractor has adopted a dynamical point of view. In contrast, we allow time to be a complex variable and look upon such solutions of the Lorenz system as analytic functions. Formal analysis gives the form and coefficients of the complex singularities of the Lorenz system. Very precise (> 500 digits) numerical computations show that the periodic orbits of the Lorenz system have singularities which obey that form exactly or very nearly so. Both formal analysis and numerical computation suggest that the mathematical analysis of the Lorenz system is a problem in analytic function theory. (Joint work with S. Sahutoglu).

We present some recent results on singular solutions of the problem of travelling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a symmetric corner of 120 degrees or a horizontal tangent at any isolated stagnation point. Moreover, the profile necessarily has a symmetric corner of 120 degrees if the vorticity is nonnegative near the free surface.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk will be in two parts.

This talk will review recent progress in understanding the effective

behavior of free boundaries in heterogeneous media.  Though motivated

by the pinning of martensitic phase boundaries, we shall explain

connections to other problems.  This talk is based on joint work with

Patrick Dondl.